A semi-Dirac point and an electromagnetic topological transition in a dielectric photonic crystal

Ying Wu

doi:10.1364/OE.22.001906

1. Introduction

Dirac cones in electron systems give rise to many intriguing transport properties, such as Klein tunneling, Zitterbewegung, and anti-localization, because the $E - k$ relation is linear at the corner of the Brillouin zone [1–3]. Similar dispersion relations, i.e., Dirac or Dirac-like cones, have also been found in classical wave systems [2–22], which lead to remarkable wave transport behaviors and interesting applications in classical waves. Representative examples are classical analogs of edge states in Quantum-Hall-Effect systems [5], extremal transmission [6,7], classical analogs of Zitterbewegung [8,9], and cloaking of an object [10].

Very recently, a semi-Dirac cone, a type of unique and unprecedented electronic band dispersion, was discovered [23] and studied [24]. It was found that, near a point in the Fermi surface in the two-dimensional (2D) Brillouin zone, the dispersion relation is linear along the symmetry line ((1,1) direction) but quadratic in the perpendicular direction [23–26]. This unique feature results in the interesting hybridized property of the associated quasiparticles that are massless along one direction, like those in graphene, but effective-mass-like along the other direction. More interestingly, it is reported that this semi-Dirac point is associated with the topological phase transition between a semi-metallic phase and a band insulator [26]. Although semi-Dirac cones in electronic systems have attracted extensive attention, to the best of my knowledge, there has been no report on their classical analogs. If the special peculiarity of a semi-Dirac cone could be transcribed into a classical system, it is possible to envisage various intriguing consequences that could be attributed to this unique dispersion relation. An obvious one is the super anisotropic wave transport behavior near the semi-Dirac point.

The electromagnetic topological transition (ETT), proposed earlier in the context of optical waves, is the electromagnetic equivalent of the “Liftshiz transition”, in which the iso-frequency surface is transformed from a closed to an open geometry as the frequency changes [27]. This transition leads to drastic changes in the nature of the electromagnetic radiation, such as an enhanced spontaneous emission rate. Recently, a strongly anisotropic metal-dielectric metamaterial was designed to demonstrate the occurrence of the optical topological transition [27]. However, metallic components bring non-negligible losses that might affect the performance of the metamaterial, and, moreover, the size of the metamaterial’s building block is much smaller than (around one twentieth of) the operating wavelength, which means that fabrication of such a material is challenging.

The semi-Dirac point and ETT seem to be two unrelated subjects. In fact, they become connected under certain circumstances. In a 2D dispersive homogeneous anisotropic medium with permittivity $ε$ and uniaxial permeability $\overset{\leftrightarrow}{μ} = d i a g (μ_{x}, μ_{y})$ , the iso-frequency surface for a transverse-electric (TE) polarized wave, i.e. $\overset{⇀}{E} = (0, 0, E_{z})$ , propagating in such a medium follows the relation

\frac{k_{x}^{2}}{μ_{y}} + \frac{k_{y}^{2}}{μ_{x}} = ω^{2} ε .

Thus, altering the sign of one component in the permeability tensor, e.g. $μ_{y}$ , would result in a transition in the topology of the iso-frequency surface. Furthermore, if the permittivity $ε$ and $μ_{y}$ are simultaneously zero at a particular frequency $ω_{0}$ , it can be shown that the dispersion relation at the center of the Brillouin zone exhibits linear-parabolic behavior near $ω_{0}$ , the peculiar yet defining property of a semi-Dirac cone. ETT is therefore intertwined with the semi-Dirac point under the condition that $ε = μ_{y} = 0$ and $μ_{x} \neq 0$ . Interestingly, this condition indicates that the material is hybridized from a zero-index material (ZIM) with zero permittivity and permeability along the x-direction and an epsilon-near-zero material (ENZM) along the y-direction. Both ZIM [10, 28–30] and ENZM [31–34] exhibit rich physics that gives rise to unconventional wave manipulation properties [35], such as super-coupling [31] and cloaking of an object [10]. A material hybridized from ZIM and ENZM is likely to spawn a large variety of possibilities in wave control. Thus, achieving a semi-Dirac point, an ETT, and hybridized properties in just one simple anisotropic material is fundamentally interesting in both theory and application.

The purpose of this work is to design a material that possesses a semi-Dirac point, supports ETT, and exhibits hybridized properties. This can be accomplished by employing accidental degeneracy in a 2D photonic crystal (PhC) with anisotropic dielectric scatterers. The advantage of using dielectric components is manifested in minimizing the loss that would hamper the functionality of the material. Numerical simulations show that there exists a semi-Dirac point with a finite frequency in the center of the Brillouin zone. At the semi-Dirac point, ETT is observed, where the topology of the iso-frequency surface is changed from an open hyperbola to a closed ellipse, and the PhC is indeed a ZIM along the x-direction and an ENZM along the y-direction. To gain deeper understanding of the semi-Dirac point, I developed a perturbation method [19,20] to investigate the dispersion relations and derived a boundary effective medium theory to study the effective medium properties. The perturbation method accurately predicts the slopes of the dispersion relation, confirming the linear-parabolic behavior of the dispersion relation in the vicinity of the semi-Dirac point. A surprising result unveiled by the method is that the linear slope decreases as $\vec{k}$ rotates away from the $Γ X$ direction and eventually vanishes when $\vec{k}$ is in the $Γ Y$ direction. The boundary effective medium theory indeed links the PhC with the special features of the semi-Dirac point and ETT to an anisotropic zero-index material. This theory offers a clear picture of the physical origin of ETT and indicates that the PhC is a hybridized material at the semi-Dirac point.

The remainder of this paper is organized as follows: the design of the PhC and its band structure, which exhibits a semi-Dirac point and ETT, are presented in Section 2. In Section 3, the perturbation method is developed and exploited to study the dispersion relations and the linear-parabolic behavior of the dispersion relation is affirmed. In Section 4, a boundary effective medium theory is deduced and ETT and the hybridized property of the PhC are subsequently discussed in the context of this theory. Two illustrative examples, beam splitting and beam shaping, are also presented in Section 4 to show the drastic change in the wave manipulation behavior induced by ETT. Conclusions are drawn in Section 5.

2. The photonic crystal system and its dispersion relation

The PhC considered in this study is a square array of elliptical cylinders with a dielectric constant of $ε_{s} = 12.5$ embedded in air ( $ε_{0} = 1$ ). The inset of Fig. 1(a) illustrates the unit cell of this PhC. The semi-minor axis of each elliptical cylinder is $r_{a} = 0.188 a$ , where $a$ is the lattice constant, and the semi-major axis is 1.3 times that of the semi-minor axis, i.e. $r_{b} = 1.3 r_{a}$ . The electromagnetic wave is transverse electric (TE) polarized and its electric field, $\overset{⇀}{E} = (0, 0, E_{z})$ , is always perpendicular to the plane of periodicity.

Fig. 1 (a) The band structure of the 2D PhC composed of a square array of elliptical dielectric cylinders. The inset shows the unit cell of the PhC. A doubly-degenerate state in the center of the Brillouin zone is found near the dimensionless frequency, 0.540, marked as “A”. In the vicinity of this point, the dispersion relation is linear along the $Γ X$ direction and quadratic along the $Γ Y$ direction, which is shown more clearly in Fig. 3(d). Near point “A”, there is another state in the center of the Brillouin zone, marked as “B”. The states at points “A” and “B” are used in the perturbation theory. The branches highlighted by black and blue dots are used to compute the effective medium parameters, which are shown in Fig. 4(a). (b) and (c) Enlarged views of the band structure for smaller and larger elliptical cylinders. The doubly-degenerate state shown in (a) splits into two single states, marked as A1 and A2, where A1 corresponds to a dipolar state and A2 corresponds to a monopolar state.

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Figure 1(a) shows the band structure of this PhC calculated by using COMSOL Multiphysics, a commercial package based on finite element simulations. There exists a doubly-degenerate point, marked as “A”, in the Brillouin zone center at the dimensionless frequency, $\tilde{ω} = ω a / 2 π c = 0.540$ , where $ω$ is the angular frequency and $c$ is the wave speed in air. This degenerate point is created by accidental degeneracy [10, 17–20] of a monopolar state (see Fig. 3(b)) and a dipolar state (see Fig. 3(c)) at the $Γ$ point, when the frequencies of these two states are deliberately tuned to be identical by adjusting the size or the material of the inclusion. Figures 1(b) and 1(c) show the band structure near the $Γ$ point for smaller and larger dielectric cylinders, whose eccentricity is kept at 1.3 but whose semi-minor axes become $0.180 a$ and $0.195 a$ , respectively. Apparent in both figures are the separated modes at the $Γ$ point marked as “A1” and “A2”, indicating a dipolar and a monopolar mode, respectively. The reversed relative positions of points A1 and A2 for smaller and larger cylinders imply that there must be a case where A1 and A2 coincide, which occurs when $r_{a} = 0.188 a$ as discussed earlier. This accidental degeneracy produces a very interesting dispersion relation in the vicinity of this degenerate point. Roughly seen in Fig. 1(a) are two linear bands, along the $Γ X$ direction, touching at Point A, and a quadratic band, along the $Γ Y$ direction, tangent to a flat band also at Point A. Below the flat band there is a directional gap in the $Γ Y$ direction.

Figures 2(a) and 2(b) present simulated three-dimensional dispersion surfaces, from different view angles, in the frequency regime from $\tilde{ω} = 0.45$ to 0.70. Two branches are obvious in these figures. The upper branch resembles the semi-Dirac cone discovered in electron systems, and the lower one is shaped like a roof, which is flat in one direction and bends down in the other directions. These two branches touch at a point, which is Point A mentioned earlier. The iso-frequency surface contours for the lower and upper branches are plotted in Figs. 2(c) and 2(d), where respective open hyperbolic and closed elliptical shapes are manifest. The change in the topology of the iso-frequency surface clearly indicates the occurrence of ETT [27] at Point A.

Fig. 2 (a) and (b) The three-dimensional band structure of the PhC. The upper surface is a semi-Dirac cone. Near its bottom, it is linear in $Δ k$ along all directions except for the $Γ Y$ direction, which is quadratic. It touches the lower surface at the Brillouin zone center near the dimensionless frequency, 0.54. The lower surface is flat in one direction and bends down along the other directions. (c) and (d) The iso-frequency surfaces of the lower and higher branches, where hyperbolic and elliptical surfaces are found, respectively.

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3. A perturbation method and the confirmation of the semi-Dirac dispersion

The unusual dispersion relation leads to several interesting questions: does the dispersion relation really behave like a semi-Dirac cone? Are the dispersion relations along the $Γ X$ direction truly linear? If yes, can their slopes be predicted? To answer these questions, I extended a perturbation theory [19,20] that I earlier developed with colleagues to study the dispersion relations, i.e. $Δ ω$ as a function of $Δ k$ , near Point A. The spirit of the method is to take a set of eigenfunctions at a particular symmetry point ( ${\overset{⇀}{k}}_{0}$ ) of interest as a basis to study the eigenstates in the vicinity of that point. Accurate prediction of the linear slope of the dispersion relation is possible because the eigenfunctions take the multiple-scattering in the PhC into account. Detailed derivations can be found in the Appendix. Here, the purpose is to identify the behavior of the dispersion relation near Point A. In previous studies on Dirac/Dirac-like cones [19,20], my colleagues and I pointed out that only the degenerate states need to be considered to evaluate the linear slopes of the dispersion relations because other far-away states contribute only to the higher order in $Δ k$ [36]. Here, it is not sufficient if I consider only the doubly-degenerate states at Point A to compute the linear slopes. The reason is that the eigenstate marked as “B” in Fig. 1(a) is very close to the doubly-degenerate states and its contribution to the linear term of the dispersion relation near Point A is not negligible. Therefore, I need to solve the following $3 \times 3$ secular equation:

\det | \begin{matrix} \frac{ω_{\overset{⇀}{k}}^{2} - ω_{B}^{2}}{c^{2}} + P_{11} & P_{12} & P_{13} \\ P_{21} & \frac{ω_{\overset{⇀}{k}}^{2} - ω_{A}^{2}}{c^{2}} + P_{22} & P_{23} \\ P_{31} & P_{32} & \frac{ω_{\overset{⇀}{k}}^{2} - ω_{A}^{2}}{c^{2}} + P_{33} \end{matrix} | = 0,

where

ω_{A}

(

ω_{B}

) is the frequency of the eigenstates at Point A (B), and

P_{i j}

represents the mode-coupling integrals among the three eigenstates whose profiles are plotted in Figs. 3(a)–3(c). Figure 3(a) shows the electric field distribution of the state located at Point B, where there is a dipolar state with its magnetic field parallel to the horizontal axis. This state is denoted as

ψ_{1}

. Figures 3(b) and 3(c) exhibit the electric field map of the doubly-degenerate states at Point A. Evident are a monopolar state (Fig. 3(b)) labeled as

ψ_{2}

and a dipolar state with its magnetic field parallel to the vertical axis (Fig. 3(c)) labeled as

ψ_{3}

. By substituting

ψ_{i} (i = 1, 2, 3)

into Eq. (8) in the Appendix, the values of

P_{i j}

are obtained to solve Eq. (2). Here, I wish to point out that the

3 \times 3

matrix can actually be downfolded to a

2 \times 2

one. This is because I am interested in the region close to Point A and I can express

ω_{\overset{⇀}{k}}

as

ω_{\overset{⇀}{k}} = ω_{A} + Δ ω_{\overset{⇀}{k}}

. Thus,

ω_{\overset{⇀}{k}}^{2} - ω_{A}^{2}

is approximated by

2 ω_{A} Δ ω_{\overset{⇀}{k}}

and

ω_{\overset{⇀}{k}}^{2} - ω_{B}^{2}

becomes a number,

ω_{A}^{2} - ω_{B}^{2}

, which does not depend on

Δ ω_{\overset{⇀}{k}}

and the dimension of the matrix in Eq. (2) can be reduced. Given that the dimensionless frequency at Point A is

{\tilde{ω}}_{A} = 0.540

, the solution to Eq. (2) is:

Δ {\tilde{ω}}_{\overset{⇀}{k}} = (\pm 0.0459 \cos β) Δ k + O (Δ k^{2}),

where

β

denotes the angle between

\overset{⇀}{k}

and the

Γ X

direction, and the dimensionless frequency,

Δ \tilde{ω} = Δ ω a / 2 π c

, is used. This solution clearly points to an important conclusion that the dispersion relation near Point A has non-zero linear dependence in

Δ k

in all directions except for the

Γ Y

direction where

\cos β

vanishes. The linear slope,

Δ {\tilde{ω}}_{\overset{⇀}{k}} / Δ k

, takes the largest value when

\overset{⇀}{k}

is along the

Γ X

symmetry axis, decreases as

\overset{⇀}{k}

rotates away from the

Γ X

direction, and vanishes when

\overset{⇀}{k}

is along the

Γ Y

symmetry axis. To validate Eq. (3), I present in Figs. 3(d) and 3(e) close-up views of the dispersion relations near Point A. The dots are the result of numerical simulations and the red lines are obtained by using Eq. (3). Excellent agreement between the dots and the lines is observed in the band structure along the

Γ X

direction, confirming the validity of the perturbation method. Along the

Γ M

direction, the lines given by Eq. (3) deviate from the numerical simulations when

\overset{⇀}{k}

is away from the

Γ

point. The main source of the deviation is the quadratic term in

Δ k

. To verify this, I performed a quadratic fit of the band structure shown in Fig. 3(e). I used

{\tilde{ω}}_{\overset{⇀}{k}} = {\tilde{ω}}_{A} + α_{1} Δ k + α_{2} Δ k^{2}

as the fitting function. The coefficients are

α_{1} = 0.0327

and

α_{2} = 0.0106

for the upper branch, and

α_{1} = - 0.0327

and

α_{2} = 0.0116

for the lower branch. The fitting results are plotted in Fig. 3(e) as dashed blue curves. It is worth mentioning that the coefficient of the linear term

α_{1}

agrees well with the result of Eq. (3), which gives

\pm 0.0324

when

β = π / 4

(

Γ M

direction), implying that the linear term of the perturbation method is accurate. The dots in Fig. 3(d) indicate that the band structure along the

Γ Y

direction does not have linear component near the doubly-degenerate point, which is predicted by Eq. (3) as well. I again performed a quadratic fit of the band structure along the

Γ Y

direction. For the upper branch, the coefficients are

α_{1} ≃ 0

and

α_{2} = 0.0481

, suggesting that the band structure is quadratic near the center of the Brillouin zone. The fitting function is sketched in Fig. 3(d) as blue dashed curves. The green solid curves in Fig. 3(d) are obtained by solving Eq. (2) to the second order in

Δ k

. While the lower green curve overlaps with the flat band, the upper one deviates from the band structure. The discrepancies between the perturbation theory and the band structure in both the

Γ M

and

Γ Y

directions are due to the fact that the contributions of the far-away states and the second-order perturbations need to be considered to determine an accurate coefficient of

Δ k^{2}

[36]. Such contributions are beyond the scope of the perturbation method described here and are therefore ignored. Figures 3(d) and 3(e) unambiguously demonstrate the linear-parabolic dispersion relation near Point A, which is further rigorously verified by the perturbation theory and the quadratic fit. In short, Point A is a semi-Dirac point.

Fig. 3 (a) The electric field pattern of the eigenstate marked as “B” in Fig. 1(a). Dark red and dark blue indicate the maximum positive and negative values, respectively. This is a dipolar state with a magnetic field parallel to the x-axis, indicated by the arrows. (b) and (c) The electric field patterns of the doubly-degenerate states marked as “A” in Fig. 1(a). A monopolar and a dipolar state with the magnetic field (arrows) perpendicular to the x-axis are evident. (d) An enlarged view of the band structure near the doubly-degenerate point. The dots are calculated by COMSOL. Linear dispersion is seen along the $Γ X$ direction, while a quadratic dispersion relation is manifest along the $Γ Y$ direction. Red solid lines and green solid curves are obtained from the perturbation theory. The blue dashed curves represent the results of quadratic fitting. (e) The same as (d) but along the $Γ M$ direction. A linear dispersion relation is seen again.

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Straightforward linear algebra reveals that the origin of the linear-parabolic dispersion relation resides in the strength of the mode-coupling integral between $ψ_{2}$ and $ψ_{3}$ , i.e. $\int_{u n i t c e l l} ψ_{2}^{*} (\overset{⇀}{r}) \nabla ψ_{3} (\overset{⇀}{r}) d \overset{⇀}{r}$ . The linear term disappears as the integral vanishes. Along the $Γ Y$ direction, this integration is zero because there is no coupling between $ψ_{2}$ and $ψ_{3}$ , as is also suggested by the symmetry of these two states. Even though accidental degeneracy is achieved, no linear dispersion is therefore found [20]. In fact, if the eigenstates are examined, it is not difficult to find that the underlying physics lies in $ψ_{3}$ , the dipolar mode of the doubly-degenerate state. As shown in Fig. 3(c), the magnetic field of this dipolar mode is polarized vertically, implying that it is a longitudinal mode along the $Γ Y$ direction but a transverse mode along the $Γ X$ direction. In electromagnetic waves, the longitudinal branch is localized and almost does not couple to the incident wave and other branches [10, 37]. Thus, a flat band associated with the longitudinal dipolar mode is found in the $Γ Y$ direction. However, since $ψ_{3}$ is transverse to the $Γ X$ direction, it easily couples to the incident wave and other branches. The coupling between $ψ_{2}$ and $ψ_{3}$ is therefore strong in the $Γ X$ direction and a linear dispersion relation is found.

4. Anisotropic effective medium theory and the electromagnetic topological transition

Earlier discussions show that it is possible to link a semi-Dirac point to ETT by proper effective medium parameters. For the PhC studied here, is the condition of proper effective medium parameters satisfied? From the band structure shown in Fig. 1(a), it is evident that the frequency of the semi-Dirac point is not low because the wavelengths in the scatterer and the background are roughly 0.52 and 1.85 times the lattice constant, respectively. The low-frequency requirements of many effective medium theories that wavelengths in the scatterer and the host are much larger than the size of the unit cell [37] seem to be contradicted. However, as long as the semi-Dirac point is located in the center of the Brillouin zone and is induced by the accidental degeneracy of a monopolar state and a dipolar state, the effective medium description may still be applicable [37]. Here, I adopt a boundary effective medium approach that was originally developed for elastic waves in [38]. The effective medium parameters are computed by using the constitutive relations, i.e.

{\bar{D}}_{z} = ε^{e f f} {\bar{E}}_{z}, and (\begin{matrix} {\bar{B}}_{x} \\ {\bar{B}}_{y} \end{matrix}) = (\begin{matrix} μ_{x}^{e f f} & 0 \\ 0 & μ_{y}^{e f f} \end{matrix}) (\begin{matrix} {\bar{H}}_{x} \\ {\bar{H}}_{y} \end{matrix}),

where the average fields are evaluated from the eigenstate fields on the boundaries of the unit cell. For example, for an eigenstate whose Bloch wave vector,

\overset{⇀}{k}

, is along the

Γ X

direction, the average electric and magnetic fields are respectively:

{\bar{E}}_{z} = (\int_{0}^{a} E_{z} (x = 0) d y + \int_{0}^{a} E_{z} (x = a) d y) / 2 a,

{\bar{H}}_{y} = (\int_{0}^{a} H_{y} (x = 0) d y + \int_{0}^{a} H_{y} (x = a) d y) / 2 a,

and the average electric displacement and magnetic induction fields are respectively:

{\bar{D}}_{z} = (\int_{0}^{a} H_{y} (x = a) d y - \int_{0}^{a} H_{y} (x = 0) d y) / (- i ω a^{2}),

{\bar{B}}_{y} = (\int_{0}^{a} E_{z} (x = a) d y - \int_{0}^{a} E_{z} (x = 0) d y) / (i ω a^{2}) .

Here, the Maxwell equation,

\nabla \times \vec{B} = \partial \vec{D} / \partial t

and

\nabla \times \vec{E} = - \partial \vec{B} / \partial t

, and Stokes’ theorem are exploited. For the eigenstates with Bloch wave vectors in the

Γ Y

direction, similar expressions can be obtained.

Figure 4(a) shows the results of the effective medium parameters evaluated by this boundary integration method using the eigenstates highlighted by the solid dots shown in Fig. 1(a). Figure 4(a) shows that the effective permittivities calculated from the eigenstates along the $Γ X$ and $Γ Y$ directions are identical, because the electric field is a scalar for TE polarized waves. However, the permeability is anisotropic owing to the vector nature of the in-plane magnetic field. Figure 4(a) shows that $μ_{y}^{e f f}$ and $ε^{e f f}$ cross zero simultaneously at the frequency of the semi-Dirac point, whereas $μ_{x}^{e f f}$ equals zero at a lower frequency $\tilde{ω} = 0.487$ . I emphasize that the effective medium theory derived here is valid near the center of the Brillouin zone.

Fig. 4 (a) Effective medium parameters evaluated with a boundary effective medium theory using the eigenstates highlighted by solid dots in Fig. 1(a). The blue triangles and black squares represent the effective permittivity $ε^{e f f}$ calculated by using the eigenstates along the $Γ Y$ and $Γ X$ directions, respectively. They almost overlap, indicating that $ε^{e f f}$ is a scalar and does not depend on the direction. The red circles represent $μ_{y}^{e f f}$ , which crosses zero simultaneously with $ε^{e f f}$ at the semi-Dirac point. The green triangles represent $μ_{x}^{e f f}$ , which crosses zero at dimensionless frequency 0.487. Note that both the blue and green triangles are missing from the frequency regime at 0.487 to 0.540, which corresponds to a band gap along the $Γ Y$ direction. No eigenstates are thus available to evaluate the related effective medium parameters. (b)-(d) The electric field for a plane wave impinging on a PhC slab in a waveguide whose walls have perfect magnetic conductor boundary conditions at the semi-Dirac frequency 0.540. Dark red and dark blue indicate the maximum positive and negative values, respectively. (b) The real part of the electric field when the incident wave is along the $Γ Y$ direction. The transmitted field is very weak. The imaginary part is orders of magnitude smaller than the real part, which is why it is not shown here. (c) and (d) The real and imaginary parts of the electric field when the incident wave is along the $Γ X$ direction. Both suggest that there is no phase change in the sample, which is a typical property of a ZIM.

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The effective medium results are indeed consistent with the ETT observed earlier. As explained earlier, the iso-frequency surface of an anisotropic material follows the relation described by Eq. (1). For the effective medium of the PhC considered here, in the frequency regime between 0.487 and 0.540, $μ_{x}^{e f f} > 0$ , $μ_{y}^{e f f} < 0$ and $ε^{e f f} < 0$ . Such a combination of signs in the effective medium parameters not only leads to hyperbolic iso-frequency surfaces, which are close to those in Fig. 2(c), but it also gives rise to a band gap along the $Γ Y$ direction and a negative band along the $Γ X$ direction. This is because $k_{y}$ is purely imaginary when $k_{x} = 0$ while $n_{x}^{e f f} = \sqrt{ε^{e f f} μ_{y}^{e f f}} < 0$ when $k_{y} = 0$ . Similar analysis can be applied to the higher branch when the frequency is above the semi-Dirac point, where all the material parameters are positive. Elliptical iso-frequency surfaces are therefore expected. The simultaneous zero $μ_{y}^{e f f}$ and $ε^{e f f}$ achieved by accidental degeneracy lead to a linear dispersion relation along the $Γ X$ direction in the vicinity of the semi-Dirac point, whereas a single zero in $ε^{e f f}$ and a positive $μ_{x}^{e f f}$ make the dispersion relation quadratic along the $Γ Y$ direction. All the behaviors predicted by the effective medium parameters are in line with the properties of the simulated band structures.

The ETT results in drastic changes in the wave manipulation properties. In Fig. 5, I demonstrate the radiation properties of a square sample with 16-by-16 rods that are illuminated by a point source located at the center of the sample at two different frequencies. Figures 5(a) and 5(b) show, respectively, the electric field and the flux distributions when the point source radiates at the dimensionless frequency 0.520, which is below the semi-Dirac point. Due to the hyperbolic shape of the dispersion relation at that frequency, the out-going wave splits into four beams, which are consistent with the iso-frequency surface. However, when the frequency is slightly above the semi-Dirac point, i.e. $\tilde{ω} = 0.544$ , the outgoing beams go mainly along the horizontal direction and the wave front is almost parallel to the vertical surfaces as shown in Fig. 5(c). The field pattern from the same source but inside a homogenous anisotropic medium is plotted in Fig. 5(d), where the effective medium parameters ( $ε^{e f f} = 0.018$ , $μ_{y}^{e f f} = 0.042$ , and $μ_{x}^{e f f} = 0.495$ ) are obtained from the effective medium theory. Almost the same field pattern is observed in Figs. 5(c) and 5(d), demonstrating the validity of the effective medium theory.

Fig. 5 A point source is placed inside the center of a square sample of 16-by-16 rods. (a) and (c) show the electric field patterns when the source frequency is below (0.520) and slightly above (0.544) the semi-Dirac point, respectively. Beam splitting and directional beam shaping are observed. (b) The radial flux as a function of the angle for the case simulated in (a). (d) The same as (c) but the sample is replaced by its effective medium. A similar pattern to that shown in (c) is found. Dark red and dark blue indicate the maximum positive and negative values, respectively.

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The effective medium theory also reveals the important hybridized feature of the semi-Dirac point studied here. Since $μ_{y}^{e f f}$ and $ε^{e f f}$ are simultaneously zero while $μ_{x}^{e f f}$ is positive, the PhC is a ZIM along the $Γ X$ direction and an ENZM along the $Γ Y$ direction. This super anisotropic characteristic is different from previously studied anisotropic zero materials, in which one component of the physical parameters is zero [39–41]. Figures 4(b)–4(d) show the electric field pattern of a TE-polarized plane wave with the frequency of the semi-Dirac point impinging on a PhC slab inside a waveguide. The waveguide has boundary walls that are perfect magnetic conductors. Clearly, when the incident wave is propagating in the $Γ X$ direction, the PhC exhibits the typical transmission property of a ZIM [10], i.e. total transmission is supported without any phase change [35] inside the material as shown in Figs. 4(c) and 4(d) for the real and imaginary parts of the electric field, respectively. However, when the PhC is illuminated by the same incident wave along the $Γ Y$ direction, the transmitted field is very weak, which is consistent with the transmission property of an ENZM. This anisotropic transport feature provides evidence that the PhC has hybridized properties of a ZIM and an ENZM.

5. Conclusions

In conclusion, I have demonstrated, by accidental degeneracy, that a 2D PhC composed of a square array of elliptical dielectric cylinders possesses a semi-Dirac point in the center of the Brillouin zone. This semi-Dirac point is associated with an ETT in its iso-frequency surface where the topology of the surface is changed from an open hyperbola to a closed ellipse. The ETT results in a drastic change in the wave manipulation behavior from beam splitting to beam shaping. A perturbation method is developed to affirm that in the vicinity of the semi-Dirac point, the dispersion relation is linear along one symmetry axis (the $Γ X$ direction) and quadratic along the perpendicular one (the $Γ Y$ direction). Furthermore, this method reveals that the linear slope decays as the Bloch wave vector rotates away from the $Γ X$ direction. An effective medium theory based on boundary integration is deduced and used to build a link between the semi-Dirac point and ETT and to reveal the hybridized properties of the PhC at the semi-Dirac point. Since the PhC is made of dielectric media, the loss would not be significant compared to if it were made of metal. The working wavelength of the semi-Dirac point is roughly twice that of the lattice constant, which is not very large and would make fabrication of such a dielectric PhC feasible.

Appendix

In this appendix, I give a detailed derivation of the perturbation method that is used in Section 2. In two dimensions, the electric field of a TE polarized wave satisfies the following wave equation:

\nabla \times (\frac{1}{μ (\vec{r})} \nabla \times E_{z} \hat{z}) = - \frac{ω^{2}}{c_{0}^{2}} ε (\vec{r}) E_{z} \hat{z},

where

ε (\vec{r})

and

μ (\vec{r})

are the permittivity and permeability, respectively, and

c_{0}

is the speed of light in air or vacuum. For periodic systems, the solution of Eq. (5) can be expressed as Bloch wave functions,

Ψ_{n \vec{k}} (\vec{r}) = u_{n \vec{k}} (\vec{r}) e^{i \vec{k} \cdot \vec{r}}

, where

u_{n \vec{k}} (\vec{r})

is a periodic function and

\vec{k}

is the Bloch wave vector. The relation between the eigenfrequency (

ω_{n \vec{k}}

) and the Bloch wave vector gives the n^th branch of the dispersion relations.

In the perturbation theory, the unperturbed Bloch states at ${\vec{k}}_{0}$ are obtained from finite element simulations, which means that $Ψ_{n {\vec{k}}_{0}} (\vec{r}) = u_{n {\vec{k}}_{0}} (\vec{r}) e^{i {\vec{k}}_{0} \cdot \vec{r}}$ is known. The Bloch states at $\vec{k}$ near ${\vec{k}}_{0}$ can be expanded as:

Ψ_{n \vec{k}} (\vec{r}) = \sum_{j} A_{n j} (\vec{k}) e^{i (\vec{k} - {\vec{k}}_{0}) \cdot \vec{r}} Ψ_{j {\vec{k}}_{0}} (\vec{r}),

where the unknown periodic function

u_{n \vec{k}} (\vec{r})

is expressed as linear combinations of

u_{n {\vec{k}}_{0}} (\vec{r})

. By substituting Eq. (6) into Eq. (5) and invoking the orthonormal properties of the Bloch wave functions, i.e.

\frac{{(2 π)}^{2}}{Ω} \int_{u n i t c e l l} Ψ_{l \vec{k}}^{*} (\vec{r}) ε (\vec{r}) Ψ_{j \vec{k}} (\vec{r}) d \vec{r} = δ_{l j}

with

Ω

and

δ_{l j}

denoting the volume of a unit cell and the Kronecker delta, respectively,

\sum_{j} [\frac{ω_{j 0}^{2} - ω_{n \vec{k}}^{2}}{c_{0}^{2}} δ_{l j} - P_{l j} (\vec{k})] A_{n j} (\vec{k}) = 0

is obtained.

Here, $P_{l j} (\vec{k})$ represents the mode-coupling integrals between two states $Ψ_{l {\vec{k}}_{0}} (\vec{r})$ and $Ψ_{j {\vec{k}}_{0}} (\vec{r})$ , and is expressed as

P_{l j} (\vec{k}) = (\vec{k} - {\vec{k}}_{0}) \cdot {\vec{p}}_{l j} - {(\vec{k} - {\vec{k}}_{0})}^{2} q_{l j},

where

{\vec{p}}_{l j} = i \frac{{(2 π)}^{2}}{Ω} \int_{u n i t c e l l} Ψ_{l {\vec{k}}_{0}}^{*} (\vec{r}) [\frac{2 \nabla Ψ_{j {\vec{k}}_{0}} (\vec{r})}{μ (\vec{r})} + (\nabla \frac{1}{μ (\vec{r})}) Ψ_{j {\vec{k}}_{0}} (\vec{r})] d \vec{r}

and

q_{l j} = \frac{{(2 π)}^{2}}{Ω} \int_{u n i t c e l l} Ψ_{l {\vec{k}}_{0}}^{*} (\vec{r}) \frac{1}{μ (\vec{r})} Ψ_{j {\vec{k}}_{0}} (\vec{r}) d \vec{r}

. Since the PhC studied in this work does not involve different magnetic permeability, i.e.

μ (\vec{r})

is 1 everywhere, the expressions for

{\vec{p}}_{l j}

and

q_{l j}

can be greatly simplified.

Acknowledgments

The author is grateful to Prof. Z.Q. Zhang, Prof. C. T. Chan, Prof. J. Li, Prof. J. Mei and Dr. X. Q. Huang for fruitful discussions. Special thanks go to Prof. P. Sheng, Prof. Y. Lai, and Prof. Z. H. Hang for their comments. I also would like to acknowledge insightful comments from anonymous reviewers, which greatly helped to improve the quality of this paper. I thank V. Unkefer for editorial work on this manuscript. This work was supported by KAUST Baseline Research Funds.

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A semi-Dirac point and an electromagnetic topological transition in a dielectric photonic crystal

Abstract

1. Introduction

2. The photonic crystal system and its dispersion relation

3. A perturbation method and the confirmation of the semi-Dirac dispersion

4. Anisotropic effective medium theory and the electromagnetic topological transition

5. Conclusions

Appendix

Acknowledgments

References and links

Cited By

Figures (5)

Equations (12)

Optics Express